Hello All,
So far I have seen many books giving good intuitive
insights into signal power and noise power.
But I am yet to grasp the significance of signals
classified as energy and how to put it to use.
For example impulse is an energy signal with energy = 1.
All finite duration aperiodic signals are energy signals.
A real valued exponential x(n) = a^n, |a|<1 is an energy
signal.
%--------------------------------------------------------------------
A bursty sinewave which occurs only once for finite duration
will be classified as power signal or energy signal?
Any pointers to good reading list is welcome.
Regards
Bharat Pathak
Arithos Designs
www.Arithos.com
significance of energy
Started by ●February 24, 2008
Reply by ●February 27, 20082008-02-27
On Feb 24, 5:46 pm, "bharat pathak" <bha...@arithos.com> wrote:> Hello All, > > So far I have seen many books giving good intuitive > insights into signal power and noise power. > > But I am yet to grasp the significance of signals > classified as energy and how to put it to use. > > For example impulse is an energy signal with energy = 1. > > All finite duration aperiodic signals are energy signals. > > A real valued exponential x(n) = a^n, |a|<1 is an energy > signal. > > %-------------------------------------------------------------------- > > A bursty sinewave which occurs only once for finite duration > will be classified as power signal or energy signal? > > Any pointers to good reading list is welcome. > > Regards > Bharat Pathak > > Arithos Designswww.Arithos.comBruel & Kjaer have a number of appnotes on the use of real instruments for energy measurement. Many are still available on the net. Try a google search for them: Bo0320.pdf Analysis of Transient and Non-stationary Signals using the Real-Time Frequency Analyzers Types 2123 and 2133 by Roger Upton Bo0438.pdf Choose your Units! (PWR, PSD, ESD) Bv0053.pdf Non-Stationary STFT Technical Review No. 1 - 2000 If you don't want real applications, never mind. Dale B. Dalrymple http://dbdimages.com
Reply by ●February 27, 20082008-02-27
>On Feb 24, 5:46 pm, "bharat pathak" <bha...@arithos.com> wrote: >> Hello All, >> >> So far I have seen many books giving good intuitive >> insights into signal power and noise power. >> >> But I am yet to grasp the significance of signals >> classified as energy and how to put it to use. >> >> For example impulse is an energy signal with energy = 1. >> >> All finite duration aperiodic signals are energy signals. >> >> A real valued exponential x(n) = a^n, |a|<1 is an energy >> signal. >> >> %-------------------------------------------------------------------- >> >> A bursty sinewave which occurs only once for finite duration >> will be classified as power signal or energy signal? >> >> Any pointers to good reading list is welcome. >> >> Regards >> Bharat Pathak >> >> Arithos Designswww.Arithos.com > >Bruel & Kjaer have a number of appnotes on the use of real instruments >for energy measurement. Many are still available on the net. Try a >google search for them: > >Bo0320.pdf >Analysis of Transient and Non-stationary Signals using the Real-Time >Frequency Analyzers Types 2123 and 2133 by Roger Upton > >Bo0438.pdf >Choose your Units! (PWR, PSD, ESD) > >Bv0053.pdf >Non-Stationary STFT >Technical Review No. 1 - 2000 > >If you don't want real applications, never mind. > >Dale B. Dalrymple >http://dbdimages.com >******************************************************** according to my experience so far, the distiction of signals to energy signals and power signals surves theoretical purposes. Specifically an energy signal has a Fourier Transform while a power signal does not have a Fourier transform, unless we introduce generalized functions, such as the Dirac d(t). Manolis C. Tsakiris
Reply by ●February 27, 20082008-02-27
On 25 Feb, 02:46, "bharat pathak" <bha...@arithos.com> wrote:> Hello All, > > � � � So far I have seen many books giving good intuitive > � � � insights into signal power and noise power. > > � � � But I am yet to grasp the significance of signals > � � � classified as energy and how to put it to use.The difference is mainly theoretical. Consider the integral 1 T lim ---- integral |x(t)|^2 dt = L T -> inf 2T -T If the limit L as T-> inf = 0, the signal x(t) is an energy signal, and if L equals some positive constant C x(t) is a power signal. In practice, this means that energy signals have some range a < t < b along the t axis where signal is 'practically' non-vanishing. Note, however, that you should not take this too literally, as e.g. a Gaussian pulse will be non-zero for all t, -inf < t < t, but these values will be 'practically' vanishing oudside a finite interval, say, [-10,10]. One consequence if th eenergy signal property is that there will be a finite number of global maxima. And because of that, it makes sense to establish a reference point on the time axis, which in turn establishes 'phase' as a useful concept. Power signals, on the other hand, have existed for all eternity and will exist forever. In which case a time reference becomes more of a diffuse concept (there are no reference points which are percieved as 'natural'), and 'phase' is not at all well-defined. Because of these technical difficulties one sees claims that 'the Fourier Transform does not exist for power signals', which is formally correct but which doesn't represent much of an obstacle in every-day life. All the useful concepts and tools of frequency domain ('frequency', 'bandwidth', 'power spectral density') are available anyway. Rune
Reply by ●February 27, 20082008-02-27
On Feb 24, 5:46 pm, "bharat pathak" <bha...@arithos.com> wrote:> Hello All, > > So far I have seen many books giving good intuitive > insights into signal power and noise power. > > But I am yet to grasp the significance of signals > classified as energy and how to put it to use....> A bursty sinewave which occurs only once for finite duration > will be classified as power signal or energy signal?The difference often has to do with whether there is a natural window or an arbitrary window for the portion of the signal of interest. For instance you can measure the entire duration of a finite signal pulse, and figure out how many feet will it lift one pound, or how many degrees will it heat up one teacup full of tea (e.g. equivalent potential energy). With a continuous signal, maybe you don't know or don't care exactly when it was turned on, or when the janitor will accidentally pull the plug and shut it off, so, in order to get home in time for dinner, you attach some unnatural or arbitrary, usually rectangular, window (some fraction of an approximation of the earth's rotation, what does that have to do with the signal?) to the signal, and measure the power per that unit window (how many teacups per second could it boil, how much horsepower could it be converted into at 100% efficiency, etc., during your arbitrary window.) IMHO. YMMV. -- rhn A.T nicholson d.0.t C-o-M
Reply by ●February 27, 20082008-02-27
Rune Allnor wrote: ...> The difference is mainly theoretical. Consider the integral > > � � � � � � 1 � � T > lim � � � ---- integral |x(t)|^2 dt � �= L > T -> inf � 2T � �-T > > If the limit L as T-> inf = 0, the signal x(t) is an > energy signal, and if L equals some positive constant C > x(t) is a power signal. > > In practice, this means that energy signals have some > range a < t < b along the t axis where signal is 'practically' > non-vanishing.You probably meant the logical complement of what you said here, no? Otherwise, the statement is trivial (every function that is not zero everywhere has a an interval where it is, well, not zero).> Note, however, that you should not take this > too literally, as e.g. a Gaussian pulse will be non-zero > for all t, -inf < t < t, but these values will be 'practically' > vanishing oudside a finite interval, say, [-10,10]. > > One consequence if th eenergy signal property is that there > will be a finite number of global maxima.I can think of an energy signal that has an infinite number of global maxima. Do I get a price?> And because of that, > it makes sense to establish a reference point on the time > axis, which in turn establishes 'phase' as a useful concept. > > Power signals, on the other hand, have existed for all eternity > and will exist forever. In which case a time reference becomes > more of a diffuse concept (there are no reference points which > are percieved as 'natural'), and 'phase' is not at all > well-defined.Are you saying that the phase of a sinusoid (being a power signal) is not well defined?> > Because of these technical difficulties one sees claims that > 'the Fourier Transform does not exist for power signals', which > is formally correct but which doesn't represent much of an > obstacle in every-day life. All the useful concepts and tools > of frequency domain ('frequency', 'bandwidth', 'power spectral > density') are available anyway.One thing I have always been curious about: what is the bandwidth of a sinusoid? If zero, then what is the bandwidth of a bandlimited periodic signal consisting of a finite number of sinusoids? Also zero? What about about the bandwidth of a general periodic function? Is bandwidth an 'additive' property?> > RuneRegards, Andor
Reply by ●February 27, 20082008-02-27
On 27 Feb, 20:54, Andor <andor.bari...@gmail.com> wrote:> Rune Allnor wrote: > > ... > > > The difference is mainly theoretical. Consider the integral > > > � � � � � � 1 � � T > > lim � � � ---- integral |x(t)|^2 dt � �= L > > T -> inf � 2T � �-T > > > If the limit L as T-> inf = 0, the signal x(t) is an > > energy signal, and if L equals some positive constant C > > x(t) is a power signal. > > > In practice, this means that energy signals have some > > range a < t < b along the t axis where signal is 'practically' > > non-vanishing. > > You probably meant the logical complement of what you said here, no? > Otherwise, the statement is trivial (every function that is not zero > everywhere has a an interval where it is, well, not zero).I meant to say exactly what I wrote. The Gaussian pulse x(t)=exp(-x^2) is non-zero fro all t, but has a vanishing integral L as above. The Gaussian is an energy signal despite of deing non-zero everywhere.> > Note, however, that you should not take this > > too literally, as e.g. a Gaussian pulse will be non-zero > > for all t, -inf < t < t, but these values will be 'practically' > > vanishing oudside a finite interval, say, [-10,10]. > > > One consequence if th eenergy signal property is that there > > will be a finite number of global maxima. > > I can think of an energy signal that has an infinite number of global > maxima. Do I get a price?You might, if you can prove that such a signal exists. But not from me.> > And because of that, > > it makes sense to establish a reference point on the time > > axis, which in turn establishes 'phase' as a useful concept. > > > Power signals, on the other hand, have existed for all eternity > > and will exist forever. In which case a time reference becomes > > more of a diffuse concept (there are no reference points which > > are percieved as 'natural'), and 'phase' is not at all > > well-defined. > > Are you saying that the phase of a sinusoid (being a power signal) is > not well defined?'Phase' is a relative measure. It makes no sense to talk about 'phase' without also defining a reference point in time. After all, one is not able to tell the difference between the cos(t) and sin(t) functions unless one sees how the maxima and zeros line up with respect to t=n*2*pi. So I would say that 'phase' is not well-defined with power signals in the sense that there are no natural time instances to select as the reference. Compare that to energy singals: In the case of the Gaussian pulse one would often choose the time of maximum as reference. In the case of a finite-duration sinusoidal it would make sense to choose the start of the non-zero segment as reference. Rune
Reply by ●February 27, 20082008-02-27
On 27 Feb., 21:14, Rune Allnor <all...@tele.ntnu.no> wrote:> On 27 Feb, 20:54, Andor <andor.bari...@gmail.com> wrote: > > > > > > > Rune Allnor wrote: > > > ... > > > > The difference is mainly theoretical. Consider the integral > > > > � � � � � � 1 � � T > > > lim � � � ---- integral |x(t)|^2 dt � �= L > > > T -> inf � 2T � �-T > > > > If the limit L as T-> inf = 0, the signal x(t) is an > > > energy signal, and if L equals some positive constant C > > > x(t) is a power signal. > > > > In practice, this means that energy signals have some > > > range a < t < b along the t axis where signal is 'practically' > > > non-vanishing. > > > You probably meant the logical complement of what you said here, no? > > Otherwise, the statement is trivial (every function that is not zero > > everywhere has a an interval where it is, well, not zero). > > I meant to say exactly what I wrote. The Gaussian pulse > x(t)=exp(-x^2) is non-zero fro all t, but has a vanishing > integral L as above. The Gaussian is an energy signal > despite of deing non-zero everywhere.I wasn't refering to the Gaussian, but to the sentence I quoted. You say "that energy signals have some range a < t < b along the t axis where signal is 'practically' non-vanishing." Power signals also have that property, ie. there exists a range a < t < b where the power signal is 'practically' non-vanishing. I would have thought that you wanted to say: "In practice, this means that for energy signals there exists a range a < t < b along the t axis outside of which the signal is 'practically' vanishing." Note the difference.> > > > Note, however, that you should not take this > > > too literally, as e.g. a Gaussian pulse will be non-zero > > > for all t, -inf < t < t, but these values will be 'practically' > > > vanishing oudside a finite interval, say, [-10,10]. > > > > One consequence if th eenergy signal property is that there > > > will be a finite number of global maxima. > > > I can think of an energy signal that has an infinite number of global > > maxima. Do I get a price? > > You might, if you can prove that such a signal exists. > But not from me.Ok, here is the proof: Imagine a series of evenly spaced rectangular unit pulses with geometrically decreasing widths. Now where do I get that price?> > > > And because of that, > > > it makes sense to establish a reference point on the time > > > axis, which in turn establishes 'phase' as a useful concept. > > > > Power signals, on the other hand, have existed for all eternity > > > and will exist forever. In which case a time reference becomes > > > more of a diffuse concept (there are no reference points which > > > are percieved as 'natural'), and 'phase' is not at all > > > well-defined. > > > Are you saying that the phase of a sinusoid (being a power signal) is > > not well defined? > > 'Phase' is a relative measure. It makes no sense to talk > about 'phase' without also defining a reference point in > time. After all, one is not able to tell the difference between > the cos(t) and sin(t) functions unless one sees how the > maxima and zeros line up with respect to t=n*2*pi. > > So I would say that 'phase' is not well-defined with power signals > in the sense that there are no natural time instances to select > as the reference.How about t0=0?> > Compare that to energy singals: In the case of the Gaussian pulse > one would often choose the time of maximum as reference. In the > case of a finite-duration sinusoidal it would make sense to choose > the start of the non-zero segment as reference. > > RuneRegards, Andor
Reply by ●February 27, 20082008-02-27
On 27 Feb, 21:33, Andor <andor.bari...@gmail.com> wrote:> On 27 Feb., 21:14, Rune Allnor <all...@tele.ntnu.no> wrote: > > > > > > > On 27 Feb, 20:54, Andor <andor.bari...@gmail.com> wrote: > > > > Rune Allnor wrote: > > > > ... > > > > > The difference is mainly theoretical. Consider the integral > > > > > � � � � � � 1 � � T > > > > lim � � � ---- integral |x(t)|^2 dt � �= L > > > > T -> inf � 2T � �-T > > > > > If the limit L as T-> inf = 0, the signal x(t) is an > > > > energy signal, and if L equals some positive constant C > > > > x(t) is a power signal. > > > > > In practice, this means that energy signals have some > > > > range a < t < b along the t axis where signal is 'practically' > > > > non-vanishing. > > > > You probably meant the logical complement of what you said here, no? > > > Otherwise, the statement is trivial (every function that is not zero > > > everywhere has a an interval where it is, well, not zero). > > > I meant to say exactly what I wrote. The Gaussian pulse > > x(t)=exp(-x^2) is non-zero fro all t, but has a vanishing > > integral L as above. The Gaussian is an energy signal > > despite of deing non-zero everywhere. > > I wasn't refering to the Gaussian, but to the sentence I quoted. You > say "that energy signals have some range a < t < b along the t axis > where signal is 'practically' non-vanishing." > > Power signals also have that property, ie. there exists a range a < t > < b where the power signal is 'practically' non-vanishing. > > I would have thought that you wanted to say: > "In practice, this means that for energy signals there exists a range > a < t < b along the t axis outside of which the signal is > 'practically' vanishing." > > Note the difference.Difference noted. Correction accepted.> > 'Phase' is a relative measure. It makes no sense to talk > > about 'phase' without also defining a reference point in > > time. After all, one is not able to tell the difference between > > the cos(t) and sin(t) functions unless one sees how the > > maxima and zeros line up with respect to t=n*2*pi. > > > So I would say that 'phase' is not well-defined with power signals > > in the sense that there are no natural time instances to select > > as the reference. > > How about t0=0?As I write this, my clock shows 27th february 2008, 21:38 and something. How does that translate relative to to t0=0? Rune
Reply by ●February 27, 20082008-02-27
On 27 Feb., 21:39, Rune Allnor <all...@tele.ntnu.no> wrote:> On 27 Feb, 21:33, Andor <andor.bari...@gmail.com> wrote: > > > > > > > On 27 Feb., 21:14, Rune Allnor <all...@tele.ntnu.no> wrote: > > > > On 27 Feb, 20:54, Andor <andor.bari...@gmail.com> wrote: > > > > > Rune Allnor wrote: > > > > > ... > > > > > > The difference is mainly theoretical. Consider the integral > > > > > > � � � � � � 1 � � T > > > > > lim � � � ---- integral |x(t)|^2 dt � �= L > > > > > T -> inf � 2T � �-T > > > > > > If the limit L as T-> inf = 0, the signal x(t) is an > > > > > energy signal, and if L equals some positive constant C > > > > > x(t) is a power signal. > > > > > > In practice, this means that energy signals have some > > > > > range a < t < b along the t axis where signal is 'practically' > > > > > non-vanishing. > > > > > You probably meant the logical complement of what you said here, no? > > > > Otherwise, the statement is trivial (every function that is not zero > > > > everywhere has a an interval where it is, well, not zero). > > > > I meant to say exactly what I wrote. The Gaussian pulse > > > x(t)=exp(-x^2) is non-zero fro all t, but has a vanishing > > > integral L as above. The Gaussian is an energy signal > > > despite of deing non-zero everywhere. > > > I wasn't refering to the Gaussian, but to the sentence I quoted. You > > say "that energy signals have some range a < t < b along the t axis > > where signal is 'practically' non-vanishing." > > > Power signals also have that property, ie. there exists a range a < t > > < b where the power signal is 'practically' non-vanishing. > > > I would have thought that you wanted to say: > > "In practice, this means that for energy signals there exists a range > > a < t < b along the t axis outside of which the signal is > > 'practically' vanishing." > > > Note the difference. > > Difference noted. Correction accepted. >hey, you snipped my signal! :-)> > > 'Phase' is a relative measure. It makes no sense to talk > > > about 'phase' without also defining a reference point in > > > time. After all, one is not able to tell the difference between > > > the cos(t) and sin(t) functions unless one sees how the > > > maxima and zeros line up with respect to t=n*2*pi. > > > > So I would say that 'phase' is not well-defined with power signals > > > in the sense that there are no natural time instances to select > > > as the reference. > > > How about t0=0? > > As I write this, my clock shows 27th february 2008, 21:38 and > something. How does that translate relative to to t0=0?Don't know. Does it matter?
